A note regarding 'random effects'

Student and the Lanarkshire milk experiment

A detailed examination of the 1930 Lanarkshire Milk Experiment (LME) by the famous statistician William Sealy Gossett ("Student"), which appeared in Biometrika in 1931, is re-examined from a more modern perspective. The LME had a complicated design whereby 67 schools in Lanarkshire were allocated to receive either raw or pasteurised milk but pupils within the schools were allocated to either receive milk or to act as controls. Student's criticisms are considered in detail and examined in terms of subsequent developments on the design and analysis of experiments, in particular as regards appropriate estimation of standard errors of treatment estimates when an incomplete blocks structure has been used. An analogy with a more modern trial in osteoarthritis is made. Suggestions are made as to how analysis might proceed if the original data were available. Some lessons for observational studies in epidemiology are drawn and it is speculated that hidden clustering structures might be an explanation as to why results may vary from observational study to observational study by more than conventionally calculated standard errors might suggest.

Extending the I-squared statistic to describe treatment effect heterogeneity in cluster, multi-centre randomized trials and individual patient data meta-analysis

Treatment effect heterogeneity is commonly investigated in meta-analyses to identify if treatment effects vary across studies. When conducting an aggregate level data meta-analysis it is common to describe the magnitude of any treatment effect heterogeneity using the I-squared statistic, which is an intuitive and easily understood concept. The effect of a treatment might also vary across clusters in a cluster randomized trial, or across centres in multi-centre randomized trial, and it can be of interest to explore this at the analysis stage. In cross-over trials and other randomized designs, in which clusters or centres are exposed to both treatment and control conditions, this treatment effect heterogeneity can be identified. Here we derive and evaluate a comparable I-squared measure to describe the magnitude of heterogeneity in treatment effects across clusters or centres in randomized trials. We further show how this methodology can be used to estimate treatment effect heterogeneity in an individual patient data meta-analysis.

A nested randomised trial of the effect of tranexamic acid on intracranial haemorrhage and infarction in traumatic brain injury (CRASH-3 trial intracranial bleeding mechanistic study): Statistical analysis plan

Background: The CRASH-3 trial is a randomised trial on the effect of tranexamic acid (TXA) versus placebo on death and disability in traumatic brain injury (TBI). The CRASH-3 intracranial bleeding mechanistic study (IBMS) is a randomised trial nested within the CRASH-3 trial to examine the effect of TXA versus placebo on intracranial bleeding and infarction. Methods: Patients eligible for the CRASH-3 trial, with a GCS of 12 or less or intracranial bleeding on a pre-randomisation CT scan are eligible for the IBMS. The occurrence of intracranial bleeding, infarction, haemorrhagic oedematous lesions, mass effect and haemorrhage evacuation is examined within 28 days of randomisation using routinely collected brain scans. The primary outcome is the volume of intra-parenchymal bleeding in patients randomised within three hours of injury (adjusted for prognostic covariates). Secondary outcomes include a composite "poor" outcome, progressive and new intracranial bleeding, intracranial bleeding after neurosurgery and cerebral infarcts seen up to 28 days post-randomisation. All outcomes will be compared between treatment groups. Statistical analyses: The primary outcome will be analysed using a covariate adjusted linear mixed model. The same analysis will be done separately for patients who undergo haemorrhage evacuation post-randomisation. We will express the effect of TXA on the composite outcome, new and progressive bleeding using relative risks and 95% CIs, and on cerebral infarcts using hazard ratios and 95% CIs. We will conduct sensitivity analyses assuming missing data are MCAR or MNAR. Conclusion: The IBMS will provide information on the mechanism of action of TXA in TBI. This pre-specified statistical analysis plan is a technical extension of the published protocol. Trial registration: The CRASH-3 trial was prospectively registered at the International Standard Randomised Controlled Trials registry (19 July 2011) and ClinicalTrials.gov (25 July 2011). The registries were updated with details for the IBMS on 20 December 2016.

Clustered allocation as a way of understanding historical controls: Components of variation and regulatory considerations

There has been increasing interest in recent years in the possibility of increasing the efficiency of clinical trials by using historical controls. There has been a general recognition that in replacing concurrent by historical controls, the potential for bias is serious and requires some down-weighting to the apparent amount of historical information available. However, such approaches have generally assumed that what is required is some modification to the standard inferential model offered by the parallel group trial. In our opinion, the correct starting point that requires modification is a trial in which treatments are allocated to clusters. This immediately shows that the amount of information available is governed not just by the number of historical patients but also by the number of centres and of historical studies. Furthermore, once one accepts that external patients may be used as controls, this raises the issue as to which patients should be used. Thus, abandoning concurrent control has implications for many aspects of design and analysis of trials, including (a) identification, pre-specification and agreement on a suitable historical dataset; (b) an agreed, enforceable and checkable plan for recruiting the experimental arm; (c) a finalised analysis plan prior to beginning the trial and (d) use of a hierarchical model with sufficient complexity. We discuss these issues and suggest approaches to design and analysis making extensive reference to the partially randomised Therapeutic Arthritis Research and Gastrointestinal Event Trial study. We also compare some Bayesian and frequentist approaches and provide some important regulatory considerations. We conclude that effective use of historical data will require considerable circumspection and discipline.

Random main effects of treatment: A case study with a network meta-analysis

If the number of treatments in a network meta-analysis is large, it may be possible and useful to model the main effect of treatment as random, that is to say as random realizations from a normal distribution of possible treatment effects. This then constitutes a third sort of random effect that may be considered in connection with such analyses. The first and most common models treatment-by-trial interaction as being random and the second, rather rarer, models the main effects of trial as being random and thus permits the recovery of intertrial information. Taking the example of a network meta-analysis of 44 similar treatments in 10 trials, we illustrate how a hierarchical approach to modeling a random main effect of treatment can be used to produce shrunk (toward the overall mean) estimates of effects for individual treatments. As a related problem, we also consider the issue of using a random-effect model for the within-trial variances from trial to trial. We provide a number of possible graphical representations of the results and discuss the advantages and disadvantages of such an approach.

Understanding Variation in Sets of N-of-1 Trials

A recent paper in this journal by Chen and Chen has used computer simulations to examine a number of approaches to analysing sets of n-of-1 trials. We have examined such designs using a more theoretical approach based on considering the purpose of analysis and the structure as regards randomisation that the design uses. We show that different purposes require different analyses and that these in turn may produce quite different results. Our approach to incorporating the randomisation employed when the purpose is to test a null hypothesis of strict equality of the treatment makes use of Nelder’s theory of general balance. However, where the purpose is to make inferences about the effects for individual patients, we show that a mixed model is needed. There are strong parallels to the difference between fixed and random effects meta-analyses and these are discussed.

The Many Flavors of Model-Based Meta-Analysis: Part I-Introduction and Landmark Data

Meta-analysis is an increasingly important aspect of drug development as companies look to benchmark their own compounds with the competition. There is scope to carry out a wide range of analyses addressing key research questions from preclinical through to postregistration. This set of tutorials will take the reader through key model-based meta-analysis (MBMA) methods with this first installment providing a general introduction before concentrating on classical and Bayesian methods for landmark data.

Mastering variation: variance components and personalised medicine

Various sources of variation in observed response in clinical trials and clinical practice are considered, and ways in which the corresponding components of variation might be estimated are discussed. Although the issues have been generally well‐covered in the statistical literature, they seem to be poorly understood in the medical literature and even the statistical literature occasionally shows some confusion. To increase understanding and communication, some simple graphical approaches to illustrating issues are proposed. It is also suggested that reducing variation in medical practice might make as big a contribution to improving health outcome as personalising its delivery according to the patient. It is concluded that the common belief that there is a strong personal element in response to treatment is not based on sound statistical evidence. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Optimal and maximin sample sizes for multicentre cost-effectiveness trials

This paper deals with the optimal sample sizes for a multicentre trial in which the cost-effectiveness of two treatments in terms of net monetary benefit is studied. A bivariate random-effects model, with the treatment-by-centre interaction effect being random and the main effect of centres fixed or random, is assumed to describe both costs and effects. The optimal sample sizes concern the number of centres and the number of individuals per centre in each of the treatment conditions. These numbers maximize the efficiency or power for given research costs or minimize the research costs at a desired level of efficiency or power. Information on model parameters and sampling costs are required to calculate these optimal sample sizes. In case of limited information on relevant model parameters, sample size formulas are derived for so-called maximin sample sizes which guarantee a power level at the lowest study costs. Four different maximin sample sizes are derived based on the signs of the lower bounds of two model parameters, with one case being worst compared to others. We numerically evaluate the efficiency of the worst case instead of using others. Finally, an expression is derived for calculating optimal and maximin sample sizes that yield sufficient power to test the cost-effectiveness of two treatments.